Question Statement

Draw the graphs of the following functions:

(i) $f(x)=e^{2x}$ (ii) $g(x)=e^{0.5x}$ (iii) $h(x)=2-e^x$ (iv) $h(x)=1+e^{-2x}$ (v) $f(x)=\ln (2x)$ (vi) $g(x)=\log (x+1)$ (vii) $h(x)=3+\log (x)$ (viii) $f(x)=e^{0.6x}$ and $g(x)=\ln (0.6x)$

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Background and Explanation

To graph exponential and logarithmic functions, we identify the base and any transformations (shifts, stretches, or reflections). We then create a table of values by substituting specific $x$-coordinates into the function to find the corresponding $y$-coordinates, which are then plotted on a Cartesian plane to form a smooth curve.

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Solution

Part (i)

The function $f(x)=e^{2x}$ is an exponential growth function. Because the coefficient in the exponent is $2$, the function grows faster than the standard $e^x$.

Part (ii)

The function $g(x)=e^{0.5x}$ is an exponential growth function. Since the coefficient $0.5$ is less than $1$, the growth is more gradual than $e^x$.

Part (iii)

The function $h(x)=2-e^x$ involves a reflection of $e^x$ across the $x$-axis followed by a vertical shift upwards by $2$ units.

Part (iv)

The function $h(x)=1+e^{-2x}$ represents exponential decay (due to the negative exponent) shifted upwards by $1$ unit. As $x$ increases, the graph approaches the horizontal asymptote $y=1$.

Part (v)

The function $f(x)=\ln (2x)$ is a natural logarithmic function. It is only defined for $x>0$. The factor of $2$ inside the log results in a horizontal compression.

Part (vi)

The function $g(x)=\log (x+1)$ is a common logarithm (base 10) shifted $1$ unit to the left. The vertical asymptote is at $x=-1$.

Part (vii)

The function $h(x)=3+\log (x)$ is a common logarithm shifted $3$ units upwards.

Part (viii)

This part compares an exponential function $f(x)=e^{0.6x}$ and a logarithmic function $g(x)=\ln (0.6x)$. Note that $g(x)$ is only defined for $x>0$.

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Key Formulas or Methods Used

• Exponential Function: $y=a\cdot e^{bx}+c$
• Logarithmic Function: $y=a\cdot {\log }_b(x-h)+k$
• Natural Logarithm: $\ln (x)={\log }_e(x)$
• Point-plotting method: Calculating $y$ for various $x$ values to determine the shape of the curve.

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Summary of Steps

1. Identify the function type: Determine if it is exponential (growth/decay) or logarithmic.
2. Create a table of values: Select a range of $x$ values (including negative, zero, and positive values where the function is defined).
3. Calculate coordinates: Use a calculator to find the corresponding $y$ values for each $x$.
4. Plot the points: Mark the $(x,y)$ pairs on a coordinate plane.
5. Sketch the curve: Connect the points with a smooth line, ensuring the graph respects asymptotes (horizontal for exponential, vertical for logarithmic).